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55 TABLE 5.6 Estimated tail index for different samples sizes. Estimated tail index for the main FX rates, gold (XAU) and silver (XAG) on different samples for both daily and 30min returns. The time intervals are measured in #time (see Chapter 6). FX rates Daily returns 30min returns Short samples 7/19786/1996 7/19886/1996 Short samples USDDEM  4.84  4.34 ±2.46  3.27 ±0.50  3.29  USDJPY  7.81  5.69 ±3.94  3.86 ±0.71  3.94  GBPUSD  4.79  4.35 ±3.02  3.37 ±0.53  3.57  USDCHF  5.24  4.15 ±2.71  3.63 ±0.55  3.61  USDFRF  4.48  4.37 ±2.85  3.52 ±0.54  3.59  USDITL  3.82  3.97 ±1.94  3.38 ±0.44  3.56  USDNLG  4.17  4.05 ±1.98  3.56 ±0.66  3.57  XAUUSD  3.65  3.88 ±2.53  4.24 ±0.99  4.00  X AGUSD  3.94  3.40 ±1.92  4.12 ±0.75  3.54 
5.4.3 Extreme Risks in Financial Markets From the practitioners point of view, one of the most interesting questions that tail studies can answer is what are the extreme movements that can be expected in financial markets? Have we already seen the largest ones or are we going to experience even larger movements? Are there theoretical processes that can model the type of fat tails that come out of our empirical analysis? The answers to such questions are essential for good risk management of financial exposures. It turns out that we can partially answer them here. Once we know the tail index, we can apply extreme value theory outside our sample to consider possible extreme movements that have not yet been observed historically. This can be achieved by a computation of the quantiles with exceedance probabilities.15 Although this chapter focuses on stylized facts, it is interesting to show an example of the application of some of these empirical studies, which is very topical to risk management. There is a debate going on to design the best hedging strategy against extreme risks. Some researchers suggest using a dynamic method by utilizing conditional distributions (McNeil and Frey, 2000). We think that for practical purposes the hedge against extreme risk must be decided on the basis of the unconditional distribution. For a large portfolio, it would be impossible to find counterparties to hedge in very turbulent states of the market. Like in the case of earthquakes, hedging this type of risk needs to be planned far in advance. Let us consider the expansion of the asymptotic cumulative distribution function from which the X, observations are drawn as F(x) = 1  " [I + b 13] (5.5) We follow here the approach developed in Dacorogna et al. (2001a).
We denote by xp and xt quantiles with respective exceedance probabilities p and t. Let n be the sample size and choose p < \/n < t; that is, xt is inside the sample, while xp is not observed. By definition (we concentrate on the positive tail), p = ax01 [l +  ] , t = ax01 [l + bx (5.6) Division of the two exceedance probabilities and rearrangement yields (5.7) Given that t is inside the sample, we can replace t by its empirical counterpart m/n, say; that is, m equals the number of order statistics Xj, which are greater than Xt. An estimator for xp is then as follows xp = Xm I (5.8) where m equals the m obtained from the tail estimation corresponding to p. To write this estimator we ignore the last factor on the righthand side of Equation 5.7. This would be entirely justified in the case of the Pareto law when b = 0. Thus xp is based on the same philosophy as the Hill estimator. For an m sufficiently small relative to n, the tails of Equation 5.5 are well approximated by those of the Pareto law, and hence Equation 5.8 is expected to do well. In fact, it is possible to prove (de Haan et al, 1994) that, for the law in Equation 5.5 (5.9) has the same limiting normal distribution as the Hill estimator. Equation 5.9 gives us a way to estimate the error of our quantile computation. Table 5.7 shows the result of a study of extreme risk using Equation 5.8 to estimate the quantiles for returns over 6 hr. This time interval is somewhat shorter than an overnight position (in #time) but is a compromise between the accuracy of the tail estimation and the length of the interval needed by risk managers. The first part of the table is produced by Monte Carlo simulations of synthetic data where the process was first fitted to the 30min returns of the USDDEM time series. The second part is the quantile estimation of the FX rates as a function of the probability of the event happening once every year, once every 5 years, and so on. Because we use here a sample of 9 years, the first two columns represent values that have been actually seen in the data set, whereas the other columns are extrapolations based on the empirically estimated tail behavior. Although the probabilities we use here seem very small, some of the extreme risks shown in Table 5.7 may be experienced by traders during their active life.
TABLE 5.7 Extreme risks in the FX market. Extreme risks over 6 hr for model distributions produced by MonteCarlo simulations of synthetic time series fitted to USDDEM, compared to empirical FX data studied through a tail estimation.    Probabilities (p)       1/10  1/15  1/20  1/25   year  year  year  year  year  year  Models:        Normal  0.4%  0.5%  0.6%  0.6%  0.7%  0.7%  Student 3  0.5%  0.8%  1.0%  1.1%  1.2%  1.2%  GARCH(U)  1.5%  2.1%  2.4%  2.6%  2.7%  2.9%  HARCH  1.8%  2.9%  3.5%  4.0%  4.3%  4.6%  USD rates:        USDDEM  1.7%  2.5%  3.0%  3.3%  3.5%  3.7%  USDJPY  1.7%  2.4%  2.9%  3.2%  3.4%  3.6%  GBPUSD  1.6%  2.3%  2.6%  2.9%  3.1%  3.2%  USDCHF  1.8%  2.7%  3.1%  3.5%  3.7%  4.0%  USDFRF  1.6%  2.3%  2.8%  3.0%  3.3%  3.4%  USDITL  1.8%  2.8%  3.4%  3.8%  4.1%  4.4%  USDNLG  1.7%  2.5%  2.9%  3.2%  3.4%  3.6%  Cross rates:        DEMJPY  1.3%  1.9%  2.2%  2.5%  2.6%  2.8%  GBPDEM  1.1%  1.7%  2.1%  2.3%  2.5%  2.6%  GBPJPY  1.6%  2.3%  2.7%  3.0%  3.2%  3.4%  DEMCHF  0.7%  1.0%  1.2%  1.3%  1.4%  1.5%  GBPFRF  1.1%  1.8%  2.2%  2.5%  2.7%  2.9% 
An interesting piece of information displayed in Table 5.7 is the comparison of empirical results and results obtained from theoretical models.16 The models parameters, including the variance of the normal and Studentr distributions, result from fitting USDDEM 30min returns. For the GARCH (1,1) model (Bollerslev, 1986), the standard maximum likelihood fitting procedure is used (Guillaume et al, 1994) and the GARCH equation is used to generate synthetic time series. The same procedure is used for the HARCH model (Miiller el al, 1997a). The models results are computed using the average m and obtained by estimating the tail index of 10 sets of synthetic data for each of the models for the aggregated time series over 6 hr. As expected, the normal distribution model fares poorly as far as the extreme risks are concerned. Surprisingly, this is also the case for 16 The theoretical processes such as GARCH and HARCH are discussed in detail in Chapter 8. Here they simply serve as examples for extreme risk estimation.
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